For any positive integer q, it is a question of Baker whether the numbers $L(1,\chi )$, where χ runs over the non‐trivial characters mod q, are linearly independent over $\mathbb{Q}$. In this work, we consider the linear independence of the $L(1,\chi )$ values over number fields where χ runs over the non‐trivial characters modulo a prime p. The dimension over $\mathbb{Q}$ and over some specific number fields is known, thanks to a result of Baker, Birch and Wirsing. But little is known about the dimension over a general number field. In this note, we study the distribution and growth of these dimensions over families of number fields not covered by the Baker–Birch–Wirsing theorem. One of the crucial ingredients is a celebrated result of Linnik about the least prime in an arithmetic progression. Some of these dimension calculations are linked to Fermat and Sophie Germain primes.

中文翻译：

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关于任意数域上的ALAN BAKER的问题

对于任何正整数q，贝克是一个问题，数字是否$\mathrm{大号}（\mathrm{1个}，\chi ）$，其中χ遍历非平凡字符mod q，在$\mathbb{问}$。在这项工作中，我们考虑了$\mathrm{大号}（\mathrm{1个}，\chi ）$数值字段上的值，其中χ以素数p为模，遍历非平凡字符。尺寸超过$\mathbb{问}$由于Baker，Birch和Wirsing的结果，在一些特定的数字字段中众所周知。但是对于一般数字域的维数知之甚少。在本说明中，我们研究了贝克－伯奇－维辛定理未涵盖的数字域族上这些维的分布和增长。关键因素之一是Linnik关于算术级数中最小素数的著名结果。其中一些尺寸计算与Fermat和Sophie Germain素数有关。